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By Jurii V. Linnik, Iosif V. Ostrovskii

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Ln+x (m = 1, 2, ... ). 's such that M(r, ip) increases more rapidly than expk r for fixed k = 1, 2, ... r). f. lfJ(t) for which M(r, lfJ) ;;;i. V(r), r ;;;i. ·J(z}=v(1)+ ~ v(n~t) n=1 zPn, n where Pn. is an increasing sequence of natural. that {v(n + 1)} t/p n. -+ 1. It is clear that this series is convergent in the whole z-plane~ and so f(z) is an entire function. Now for x > 0 we have f(x) ;;;i. v(x); indeed, setting n = [x], we see that f (x} > v (n + 1) n-PnxPn > v (n + 1) > v (x). , and = f (er}/j (1) > v (er}// (1) = V 2 (r}lf (1) > V (r), r > ro.

D) Since B( fl) is convex, we have B ("11) ~ B ("12) and the assertion follows. + B' ("12) ("11 - "12), a < "111 "12 < b, II. 'S COROLLARY 1. If I{}(t) is a normalized ridge function in the strip a < l{}(i11) > 0 for a< Tl < b. 2 (b), argl{}(ifl) = const = arg 1()(0) = 0 (mod 211'), a< 11 < b. COROLLARY 2. Every entire normalized ridge function l{}(t) having no zeros is of the form l{}(t) = ef, where f(t) is an entire function taking real values on the imaginary axis, f(O) = 0. 4 the function l{}(t) may be expressed as l{}(t) = ef(t), where f (t) is an entire function, f (0) = 0.

L{)(t; F) is analytic in the half-plane Im t > 0 {Im t < 0) with a finite value of h+(F) (h_(F)) if and only if Iext F (rext F) is finite. f. 'P(t; F) has rate of growth exceeding order 1 and is of normal type, then Wp(x) > 0 for all positive x. ft; (x)) . (F) -_ 1im • lnx X-+oo If K = K(F) is finite, a more discriminating· description of the rate of decrease of Wp(x) is given by "- (F) = lim x-x In+ (1/W F (x)). The necessary and sufficient condition established at the beginning of this section for 'P(t; F) to be an entire function is clearly equivalent to the following: K(F) ;;;;..

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